A semi-uniform-price auction for multiple objects

Abstract

This paper proposes a semi-uniform payment rule for selling multiple homogeneous objects. Under the proposed auction, all bidders pay a uniform price equal to the highest losing bid, except the bidder with the highest losing bid who, under some circumstances, pays the second highest losing bid. We show that bidders in this auction face an incentive, on the margin, to increase their bids vis-a-vis their bids in a uniform-price auction. This incentive is sufficient to eliminate the zero revenue equilibrium that has been identified in the multiple-object, uniform-price auction literature.

Appendix

Proof of Proposition 1

First, rewrite Eq. (7) as follows:

$$\begin{aligned} \frac{\partial \Pi ^{su}}{\partial b_{2}}=h_{1}(b_{2})\big (v_{2}^{i} -2b_{2} +\tilde{c}_2^i \big ) + b_{2}h_{1}(b_{2})\Theta . \end{aligned}$$

(13)

For simplicity, we omit the term i in what follows. Differentiating with respect to $b_2$ gives the second-order condition as follows:

$$\begin{aligned} \frac{\partial ^2 \Pi ^{su}}{\partial b_{2}^2}= & {} h_{1}(b_{2})\big ( -2 +\tilde{c}'_2 \big ) +h'_{1}(b_{2})\big ( v_{2}-2b +\tilde{c}_2 \big ) + h_{1}(b_{2})\Theta + b_{2}h'_{1}(b_{2})\Theta \nonumber \\&+ b_{2}h_{1}(b_{2})\Theta '. \end{aligned}$$

(14)

After some manipulations, we have the following:

$$\begin{aligned} \frac{\partial ^2 \Pi ^{su}}{\partial b_{2}^2}=h_{1}(b_{2})\big ( -2 +\tilde{c}'_2 +\Theta +b_2\Theta ' \big ) +h'_{1}(b_{2})\big ( v_{2}-2b +\tilde{c}_2 +b_2\Theta \big ).\quad \end{aligned}$$

(15)

To ensure that the bidding function for the second unit is a local maxima, we need to show that the above expression is negative in the neighborhood of the optimal bid given by the FOC. The first part of (15) is strictly negative, because $\tilde{c}_2'<0$, $\Theta \le 1$, and $\Theta '<0$. Note that it is straightforward to check that $\Theta '<0$ given that $H_1$ stochastically dominates $H_2$. The second part of (15) is equal to zero in the neighborhood of $b_2^*$. It follows that $b_2^*$ is a local maxima. $\square$

Proof of Proposition 2

Denote by $p^*$ the equilibrium clearing price in the uniform-price auction. According to Engelbrecht-Wiggans and Kahn (1998) and Khezr and Menezes (2017), when the number of bidders is at least as large as the number of units, that is, $N\ge Q$, all bidders bid zero for the second unit in the uniform-price auction, irrespective of their values. Denote by ${v_1}_{(Q+1)}$ the $Q+1\hbox {th}$ highest value for the first unit. Since $N\ge Q$ and it is a weakly dominant strategy for bidders to bid their value for the firs unit, then in the uniform-price auction, $p^*$ is either equal to zero or equal to the ${v_1}_{(Q+1)}$. In fact, in the case where the number of bidders is exactly equal to the number of units, the Qth highest bid is equal to the lowest value for the first unit among the bidders. Therefore, the clearing price, which is equal to the $Q+1$th highest bid, becomes zero: all bidders bid zero for the second unit, so the aggregation of bids results in Q positive bids which are the values for the first units and Q zero bids, which are bids for the second unit. Further note that when $N>Q$, then the aggregation of bids results in N positive bids where the $Q+1$th highest bid among them sets the clearing price.

In the semi-uniform-price auction, there are two possibilities. Define $\tilde{v}_2$ as the highest realized value for the second unit among bidders. First, if $b_2(\tilde{v})$ is larger than ${v_1}_{(Q+1)}$, then the semi-uniform-price auction results in a higher revenue for sure. To see this, imagine the worst case scenario where only $b_2(\tilde{v})>{v_1}_{(Q+1)}$ and all other values for the second units are such that $b_2(v)<{v_1}_{(Q+1)}$. In this case, the auction clearing price is greater than ${v_1}_{(Q+1)}$ and the bidder who has $\tilde{v}_2$ pays ${v_1}_{(Q+1)}$. Therefore, the overall auction revenue is strictly higher than the one for the uniform-price auction. Second, is when $b_2(\tilde{v})$ is smaller than ${v_1}_{(Q+1)}$. In this case, the two auctions become equivalent. Thus, the semi-uniform-price auction has a higher expected payoff than the uniform-price auction.

The proof for efficiency follows a similar argument. A mechanism is more efficient as long as it allocates the objects to bidders with higher values. Of course, here, we are analyzing a second-best case and neither of the two mechanisms are fully efficient, that is, always allocate the objects to the buyers with the highest values.

In the first case identified above, the semi-uniform auction is more efficient, because there is a higher chance that it allocates objects to bidders with higher values. This is because in the uniform-price auction, all bidders bid zero for the second unit, and if, for instance, there exist a bidder, such that her value for the second unit is larger than at least the value of some bidders for the first unit, she will never receive a second object. However, given that $b_2>0$ for the same bidder in the semi-uniform-price auction, there is a positive probability that this bidder receives a second unit. In the second case, where $N>Q$, both auctions allocate the object in the same way. Thus, we can conclude that the semi-uniform-price auction would allocate the objects more efficiently compared to the uniform-price auction. $\square$